Chapter 9.5, Problem 92E

a.

To determine

Find a cubic model for the data.

Answer to Problem 92E

  $\boxed{y=-0.0157t^3+0.2766t^2-1.008t+9.6326}$

Explanation of Solution

Given information:

The table shows the average prices $f(t)$ (in cents per kilowatt hour) of residential electricity in the United States from 2003 through 2010. (Source: U.S. Energy Information Administration)

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Use the regression feature of a graphing utility to find a cubic model for the data. Let $t$ represent the year, with $t=3$ corresponding to 2003

Calculation:

Consider the below table

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Here $t=3$ corresponds to the year $2003$

First enter the data in calculator

For this press the button $STAT$ and select $EDIT$ and press Enter

Enter years in $L_1$ and price in $L_2$

  

To find the cubic model use linear regression in calculator

Press the button $STAT$ and select $CALC$ and select the option $6$ and press Enter,Enter

Hence, the cubic model is $\boxed{y=-0.0157t^3+0.2766t^2-1.008t+9.6326}$

  

b.

To determine

Plot the data and the model in the same viewing window.

Answer to Problem 92E

  

Explanation of Solution

Given information:

The table shows the average prices $f(t)$ (in cents per kilowatt hour) of residential electricity in the United States from 2003 through 2010. (Source: U.S. Energy Information Administration)

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Use the graphing utility to plot the data and the model in the same viewing window.

Calculation:

Consider the below table

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Data and model are shown below

Here window settings are $[0,10,2]$ by $[5,12,2]$

  

c.

To determine

Use binomial coefficients to write in $g(t)$ standard form.

Answer to Problem 92E

  $\boxed{g(t)=-0.157t^3+0.0411t^2+0.5805t+9.5451}$

Explanation of Solution

Given information:

The table shows the average prices $f(t)$ (in cents per kilowatt hour) of residential electricity in the United States from 2003 through 2010. (Source: U.S. Energy Information Administration)

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

You want to adjust the model so that $t=3$ corresponds to $2008$ rather than $2003$ . To do this, you shift the graph of five units to the left to obtain $g(t)=f(t+5)$ .Use binomial coefficients to write in $g(t)$ standard form.

Calculation:

Consider the below table

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Consider the model $g(t)=f(t+5)$

  $\begin{array}{l}g\left(t\right)=-0.0157{\left(t+5\right)}^3+0.2766{\left(t+5\right)}^2-1.008\left(t+5\right)+9.6326\\ =-0.0157\left(t^3+3t^2\times 5+3t\times 5^2+5^3\right)+0.2766(t^2+2\times t\times 5+5^2)\\ -1.008\left(t+5\right)+9.6326\end{array}$

  $\begin{array}{l}=-0.0157t^3+t^2\left(-0.0157\times 15+0.2766\right)+t\left(-0.0157\times 75+2.766-1.008\right)\\ +\left(9.6326-0.0157\times 125+25\times 0.276-1.008\times 5\right)\\ =-0.0157t^3+0.0411t^2+0.5805t+9.5451\end{array}$

Hence the required function is $\boxed{g(t)=-0.157t^3+0.0411t^2+0.5805t+9.5451}$

d.

To determine

Plot the graph $g$ in the same viewing window as $f$ .

Answer to Problem 92E

  

Explanation of Solution

Given information:

The table shows the average prices $f(t)$ (in cents per kilowatt hour) of residential electricity in the United States from 2003 through 2010. (Source: U.S. Energy Information Administration)

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Use the graphing utility to graph $g$ in the same viewing window as $f$ .

Calculation:

Consider the below table

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Consider the graph of $g(t)$ in the above viewing window

  

e.

To determine

Use both models to predict the average price in 2011.

Answer to Problem 92E

  $\boxed{\text{11}.\text{17 with f}\left(\text{t}\right),\text{11}.\text{51 with g}\left(\text{t}\right)}$

Explanation of Solution

Given information:

The table shows the average prices $f(t)$ (in cents per kilowatt hour) of residential electricity in the United States from 2003 through 2010. (Source: U.S. Energy Information Administration)

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Use both models to predict the average price in 2011. Do you obtain the same answer?

Calculation:

Consider the below table

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Evaluate $f(t)$ at $t=11$ and $g(t)$ at $t=5$ to predict the price in $2011$

Hence average price in $2011$ is $\boxed{\text{11}.\text{17 with f}\left(\text{t}\right),\text{11}.\text{51 with g}\left(\text{t}\right)}$

  

f.

To determine

Explain that part (e) is reasonable or not.

Answer to Problem 92E

The answer is not reasonable.

Explanation of Solution

Given information:

The table shows the average prices $f(t)$ (in cents per kilowatt hour) of residential electricity in the United States from 2003 through 2010. (Source: U.S. Energy Information Administration)

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Do your answers to part (e) seem reasonable? Explain.

Calculation:

Consider the below table

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Hence, the answer is not reasonable because prices are gradually increasing from the table but here it seems to be decreasing.

g.

To determine

Find the factor to change in the average price.

Answer to Problem 92E

Time is the main factor that change the price.

Explanation of Solution

Given information:

The table shows the average prices $f(t)$ (in cents per kilowatt hour) of residential electricity in the United States from 2003 through 2010. (Source: U.S. Energy Information Administration)

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

What factors do you think may have contributed to the change in the average price?

Calculation:

Consider the below table

  $\begin{array}{cc}Year& Average,price,f(t)\\ \text{2003}& 8.72\\ 2004& 8.95\\ 2005& 9.45\\ 2006& 10.40\\ 2007& 10.65\\ 2008& 11.26\\ 2009& 11.51\\ 2010& 11.58\end{array}$

Hence, the time is the main factor that change the price.